I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's Cours d'Analyse).

I also know that mathematicians in the early nineteenth century had some false beliefs about real analysis which were later uprooted as the process of rigorization continued. For example, it was widely believed that a continuous real function must be differentiable except at isolated points - a claim which Weierstrass refuted in the 1870s by defining a function that is continuous everywhere but differentiable nowhere. (http://en.wikipedia.org/wiki/Weierstrass_function).

What I'd like to know is whether at any point an "axiom" was proposed for real arithmetic, which subsequently turned out to be false.

I'd also like to hear about such cases from other branches of mathematics.

However, I'm not so interested in cases of inconsistent sets of axioms (e.g. Gottlob Frege's Grundgesetze, or Church's first formulation of the lambda calculus).

The best example I've found so far is Leibniz's "principle of continuity", according to which "what is true up to the limit is true at the limit". Apparently this was sometimes called an "axiom" although it is obviously not true in general. (I'm getting this from Lakatos's Proofs and Refutations, pg. 128). I'm not entirely happy with this example, because it's so obviously false that I can't believe that it was accepted, except as a heuristic.

Statements like that principle of continuity were accepted because people had an idea of what they meant by «property» that made it hold. It is in that vein that people believed that all functions were smooth, say: of course Euler and friends knew about the absolute value! but this did not interfere, in their minds, because they managed to exclude such beasts. Weierstrass was preceded by a rethinking of what a function is.
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Mariano Suárez-Alvarez♦Dec 19 '12 at 3:49

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Since this question is asking for a list, it should be community wiki.
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HJRWDec 19 '12 at 6:39

I agree with HW. I added the big-list tag, and flagged for moderator attention to make it CW.
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Emil JeřábekDec 19 '12 at 17:00

What is the difference between "a set of axioms, one of which subsequently turned out to be false" and "an inconsistent set of axioms"?
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ogicu8abruokDec 15 '13 at 14:57

4 Answers
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Another example from real analysis would be the question of the pointwise convergence of the Fourier series of a continuous function (defined on a closed interval). Many people, including Dirichlet and even the master rigorist Weierstrass himself, believed that the Fourier series of such a function converges pointwise everywhere to the function itself. Some clung on to this belief so strongly that they even viewed it as an infallible axiom.

Hence, one can imagine the great upset when, in 1876, Paul du Bois-Reymond proved the existence of a continuous function whose Fourier series diverges at a point. His proof is non-constructive and uses a method called the principle of condensation of singularities. I have absolutely no idea how the method works, but I do know of a very common proof that uses the Baire Category Theorem (using the Baire Category Theorem, one can also prove the existence of continuous functions that are not differentiable at any point).

After the dust had settled in the wake of du Bois-Reymond's seismic discovery, people started fervently believing that there should exist a continuous function whose Fourier series diverges everywhere - an opinion that lay on the other extreme! Andrei Kolmogorov inadvertently lent support to this claim by exhibiting, in 1926, an $ {L^{1}}([- \pi,\pi]) $-function whose Fourier series diverges everywhere. However, there was great upheaval once more in Fourier-land when the combined efforts of Lennart Carleson and Richard Hunt in the late 1960's showed that the Fourier series of any $ f \in {L^{p}}([- \pi,\pi]) $ converges almost everywhere to $ f $, for all $ p > 1 $ (this result subsumes the case of continuous functions). During an interview with the AMS, Carleson revealed that he had originally tried to disprove his result (pertaining to $ p = 2 $), but in the end, his failure to produce a counterexample convinced him that he should be working in the other direction instead.

Therefore, in the field of Fourier analysis, viewpoints have changed and cherished beliefs have been destroyed - twice.

There are things that were believed, until later disproved. And even things with proofs later found to be faulty. But these are not things stated as axioms, which is what the OP curiously asks for.
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Gerald EdgarDec 19 '12 at 14:15

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"Condensation of Singularities" was invented by Hermann Hankel. Basically it is the following: You have a function $f$ that has a singularity at 0. Make a periodic function $g$ from it that has this singularity at every integer. Then form the sum $G(x) = \sum_{n=0}^\infty a_n g(n x)$, with the $a_n$ chosen such that the sum converges. Then $G$ has this singluarity at every rational point. This can be done with mny types of singularities, provided that they are well-behaved enough under addition.
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remDec 22 '12 at 13:53

One of the most well-known examples is Frege's axiom of general comprehension, which asserts that for any definite property $P$, one may form the set $$\{x\mid\ P(x)\ \},$$ consisting of all objects $x$ with property $P$. This natural-seeming principle is one of the main axioms of what is now known as naive set theory, and formed a central axiom in Frege's Begriffsschrift and later the Grundgezetze (which I see now that you said you weren't interested in, oh well), intended as a formal logical foundation of arithmetic and all mathematics. But the axiom was famously refuted by Betrand Russell with the Russell paradox, showing that there can be no set $R=\{x\mid\ x\notin x\ \}$, consisting of the sets $x$ that are not members of themselves, since then $R\in R\iff R\notin R$, a contradiction.

And for another example of this type, recently my student Norman Perlmutter has refuted the existence of what have been called hypercompact cardinals (but which we now call them strongly hypercompact cardinals).
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Joel David HamkinsDec 19 '12 at 1:01

Correction: the inconsistent version of hypercompact cardinals are now called excessively hypercompact. So Norman has proved in a section of his dissertation that there are no excessively hypercompact cardinals. Meanwhile, what are now called the hypercompact cardinals, after a correction in light of Norman's observation, have a consistency strength below the Woodin-for-supercompact cardinals, which he has proved are the same as the Vopenka cardinals.
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Joel David HamkinsDec 20 '12 at 22:19

In the mathematical theory of social welfare, it is possible to create a list of axioms that lead to a contradiction. For example, in voting theory, the following axioms for a voting system are considered reasonable in order for the system to qualify as being fair:

Each voter can have any set of rational preferences. This requirement is called “universal admissibility”.

If a voter prefers Candidate A to Candidate B, and Candidate B to Candidate C, then he/she prefers A to C. This requirement is called “transitivity”.

If every voter prefers A to B, then the group prefers A to B. This is sometimes called the “unanimity” condition.

If every voter prefers A to B, then any change in preferences that does not affect this relationship must not affect the group preference for A over B. For example, if a set of historians unanimously decides that Abraham Lincoln was a better president than Chester A. Arthur, a changing opinion of Bill Clinton should not affect this decision. This more subtle requirement is called “independence from irrelevant alternatives”.

There are no dictators. In other words, no voter exists whose preferences determine the preferences of the whole group.

The mathematical economist Kenneth Arrow showed in a landmark paper (stemming from his PhD thesis) that one obtains a contradiction if all five assumptions are assumed to hold. In fact, Assumptions (1) - (4) imply the existence of a dictator. However, these assumptions seem fairly reasonable and consistent, so the fact that they are contradictory is why Arrow named his paper “A Difficulty in the Concept of Social Welfare”. His result is known nowadays as Arrow's Impossibility Theorem.

I'm sorry - but how does that answer the question?
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Felix GoldbergDec 19 '12 at 12:00

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As stated here, axiom 4 seems to be an immediate consequence of axiom 3. I suspect that axiom 4 should state that, if the group prefers A to B (not necessarily as a result of unanimity), then it continues to do so when irrelevant alternatives are changed.
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Andreas BlassDec 19 '12 at 12:52

Perhaps one of the earliest examples would be with the Pythagoreans, who held that any two magnitudes were commensurable, measured as integer multiples of a smaller common unit, a belief that was connected with their mystical religious views and also with their mathematical theory of musical harmony. The Pythagoreans were shocked by the discovery of incommensurable numbers, such as $\sqrt{2}$.

But it may be anachronistic to refer to the fundamental Pythagorean beliefs or principles as "axioms".

I think the point about possible anachronisms is a good one. The meaning of "axiom" has changed over the centuries, which makes the question slightly hard to interpret if we take examples from anything more than about a century ago.
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Tom LeinsterDec 19 '12 at 23:25

Tom, I basically agree with that, except that I think we would also both have no problem referring to the Euclidean axioms.
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Joel David HamkinsDec 20 '12 at 0:21